LMFDB - Embedded newform 2151.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.22231$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22231 q^{2} -0.505952 q^{4} +0.403117 q^{5} -4.45690 q^{7} -3.06306 q^{8} +O(q^{10})$	\(q+1.22231 q^{2} -0.505952 q^{4} +0.403117 q^{5} -4.45690 q^{7} -3.06306 q^{8} +0.492736 q^{10} -2.17391 q^{11} +5.85396 q^{13} -5.44773 q^{14} -2.73211 q^{16} +7.98275 q^{17} +2.00460 q^{19} -0.203958 q^{20} -2.65720 q^{22} +0.679568 q^{23} -4.83750 q^{25} +7.15536 q^{26} +2.25498 q^{28} -5.44529 q^{29} -0.595692 q^{31} +2.78662 q^{32} +9.75742 q^{34} -1.79666 q^{35} +5.86330 q^{37} +2.45025 q^{38} -1.23477 q^{40} +0.396926 q^{41} +12.0393 q^{43} +1.09990 q^{44} +0.830645 q^{46} -6.38497 q^{47} +12.8640 q^{49} -5.91293 q^{50} -2.96182 q^{52} +9.69118 q^{53} -0.876342 q^{55} +13.6518 q^{56} -6.65585 q^{58} +2.87181 q^{59} -2.31211 q^{61} -0.728121 q^{62} +8.87034 q^{64} +2.35983 q^{65} +11.4712 q^{67} -4.03889 q^{68} -2.19608 q^{70} +11.4938 q^{71} +7.74678 q^{73} +7.16679 q^{74} -1.01423 q^{76} +9.68892 q^{77} -14.2836 q^{79} -1.10136 q^{80} +0.485167 q^{82} +6.99235 q^{83} +3.21799 q^{85} +14.7158 q^{86} +6.65882 q^{88} +16.1528 q^{89} -26.0905 q^{91} -0.343829 q^{92} -7.80444 q^{94} +0.808091 q^{95} -8.60226 q^{97} +15.7238 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.22231	0.864306	0.432153	−	0.901800i	$-0.357754\pi$
$2$	1.22231	0.864306	0.432153	+	0.901800i	$0.357754\pi$
$3$	0	0
$3$	0	0
$4$	−0.505952	−0.252976
$5$	0.403117	0.180280	0.0901398	−	0.995929i	$-0.471269\pi$
$5$	0.403117	0.180280	0.0901398	+	0.995929i	$0.471269\pi$
$6$	0	0
$7$	−4.45690	−1.68455	−0.842276	−	0.539047i	$-0.818785\pi$
$7$	−4.45690	−1.68455	−0.842276	+	0.539047i	$0.818785\pi$
$8$	−3.06306	−1.08295
$9$	0	0
$10$	0.492736	0.155817
$11$	−2.17391	−0.655460	−0.327730	−	0.944772i	$-0.606284\pi$
$11$	−2.17391	−0.655460	−0.327730	+	0.944772i	$0.606284\pi$
$12$	0	0
$13$	5.85396	1.62360	0.811798	−	0.583939i	$-0.198489\pi$
$13$	5.85396	1.62360	0.811798	+	0.583939i	$0.198489\pi$
$14$	−5.44773	−1.45597
$15$	0	0
$16$	−2.73211	−0.683027
$17$	7.98275	1.93610	0.968051	−	0.250753i	$-0.0806781\pi$
$17$	7.98275	1.93610	0.968051	+	0.250753i	$0.0806781\pi$
$18$	0	0
$19$	2.00460	0.459888	0.229944	−	0.973204i	$-0.426146\pi$
$19$	2.00460	0.459888	0.229944	+	0.973204i	$0.426146\pi$
$20$	−0.203958	−0.0456064
$21$	0	0
$22$	−2.65720	−0.566517
$23$	0.679568	0.141700	0.0708499	−	0.997487i	$-0.477429\pi$
$23$	0.679568	0.141700	0.0708499	+	0.997487i	$0.477429\pi$
$24$	0	0
$25$	−4.83750	−0.967499
$26$	7.15536	1.40328
$27$	0	0
$28$	2.25498	0.426151
$29$	−5.44529	−1.01116	−0.505582	−	0.862778i	$-0.668722\pi$
$29$	−5.44529	−1.01116	−0.505582	+	0.862778i	$0.668722\pi$
$30$	0	0
$31$	−0.595692	−0.106989	−0.0534947	−	0.998568i	$-0.517036\pi$
$31$	−0.595692	−0.106989	−0.0534947	+	0.998568i	$0.517036\pi$
$32$	2.78662	0.492610
$33$	0	0
$34$	9.75742	1.67338
$35$	−1.79666	−0.303690
$36$	0	0
$37$	5.86330	0.963921	0.481960	−	0.876193i	$-0.339925\pi$
$37$	5.86330	0.963921	0.481960	+	0.876193i	$0.339925\pi$
$38$	2.45025	0.397483
$39$	0	0
$40$	−1.23477	−0.195235
$41$	0.396926	0.0619894	0.0309947	−	0.999520i	$-0.490133\pi$
$41$	0.396926	0.0619894	0.0309947	+	0.999520i	$0.490133\pi$
$42$	0	0
$43$	12.0393	1.83598	0.917988	−	0.396609i	$-0.129813\pi$
$43$	12.0393	1.83598	0.917988	+	0.396609i	$0.129813\pi$
$44$	1.09990	0.165815
$45$	0	0
$46$	0.830645	0.122472
$47$	−6.38497	−0.931344	−0.465672	−	0.884957i	$-0.654187\pi$
$47$	−6.38497	−0.931344	−0.465672	+	0.884957i	$0.654187\pi$
$48$	0	0
$49$	12.8640	1.83771
$50$	−5.91293	−0.836215
$51$	0	0
$52$	−2.96182	−0.410730
$53$	9.69118	1.33119	0.665593	−	0.746315i	$-0.268179\pi$
$53$	9.69118	1.33119	0.665593	+	0.746315i	$0.268179\pi$
$54$	0	0
$55$	−0.876342	−0.118166
$56$	13.6518	1.82429
$57$	0	0
$58$	−6.65585	−0.873955
$59$	2.87181	0.373878	0.186939	−	0.982372i	$-0.440143\pi$
$59$	2.87181	0.373878	0.186939	+	0.982372i	$0.440143\pi$
$60$	0	0
$61$	−2.31211	−0.296036	−0.148018	−	0.988985i	$-0.547289\pi$
$61$	−2.31211	−0.296036	−0.148018	+	0.988985i	$0.547289\pi$
$62$	−0.728121	−0.0924715
$63$	0	0
$64$	8.87034	1.10879
$65$	2.35983	0.292701
$66$	0	0
$67$	11.4712	1.40143	0.700715	−	0.713442i	$-0.252865\pi$
$67$	11.4712	1.40143	0.700715	+	0.713442i	$0.252865\pi$
$68$	−4.03889	−0.489787
$69$	0	0
$70$	−2.19608	−0.262481
$71$	11.4938	1.36406	0.682030	−	0.731324i	$-0.261097\pi$
$71$	11.4938	1.36406	0.682030	+	0.731324i	$0.261097\pi$
$72$	0	0
$73$	7.74678	0.906692	0.453346	−	0.891335i	$-0.350230\pi$
$73$	7.74678	0.906692	0.453346	+	0.891335i	$0.350230\pi$
$74$	7.16679	0.833122
$75$	0	0
$76$	−1.01423	−0.116340
$77$	9.68892	1.10416
$78$	0	0
$79$	−14.2836	−1.60703	−0.803514	−	0.595285i	$-0.797039\pi$
$79$	−14.2836	−1.60703	−0.803514	+	0.595285i	$0.797039\pi$
$80$	−1.10136	−0.123136
$81$	0	0
$82$	0.485167	0.0535778
$83$	6.99235	0.767510	0.383755	−	0.923435i	$-0.374631\pi$
$83$	6.99235	0.767510	0.383755	+	0.923435i	$0.374631\pi$
$84$	0	0
$85$	3.21799	0.349040
$86$	14.7158	1.58684
$87$	0	0
$88$	6.65882	0.709833
$89$	16.1528	1.71219	0.856096	−	0.516818i	$-0.172883\pi$
$89$	16.1528	1.71219	0.856096	+	0.516818i	$0.172883\pi$
$90$	0	0
$91$	−26.0905	−2.73503
$92$	−0.343829	−0.0358466
$93$	0	0
$94$	−7.80444	−0.804966
$95$	0.808091	0.0829084
$96$	0	0
$97$	−8.60226	−0.873427	−0.436714	−	0.899601i	$-0.643858\pi$
$97$	−8.60226	−0.873427	−0.436714	+	0.899601i	$0.643858\pi$
$98$	15.7238	1.58835
$99$	0	0
$100$	2.44754	0.244754
$101$	11.0688	1.10139	0.550693	−	0.834708i	$-0.314364\pi$
$101$	11.0688	1.10139	0.550693	+	0.834708i	$0.314364\pi$
$102$	0	0
$103$	−5.71739	−0.563351	−0.281675	−	0.959510i	$-0.590890\pi$
$103$	−5.71739	−0.563351	−0.281675	+	0.959510i	$0.590890\pi$
$104$	−17.9310	−1.75828
$105$	0	0
$106$	11.8456	1.15055
$107$	13.9647	1.35002	0.675011	−	0.737808i	$-0.264139\pi$
$107$	13.9647	1.35002	0.675011	+	0.737808i	$0.264139\pi$
$108$	0	0
$109$	−7.04998	−0.675265	−0.337633	−	0.941278i	$-0.609626\pi$
$109$	−7.04998	−0.675265	−0.337633	+	0.941278i	$0.609626\pi$
$110$	−1.07116	−0.102132
$111$	0	0
$112$	12.1767	1.15059
$113$	−17.5817	−1.65394	−0.826972	−	0.562243i	$-0.809938\pi$
$113$	−17.5817	−1.65394	−0.826972	+	0.562243i	$0.809938\pi$
$114$	0	0
$115$	0.273946	0.0255456
$116$	2.75505	0.255800
$117$	0	0
$118$	3.51025	0.323144
$119$	−35.5784	−3.26146
$120$	0	0
$121$	−6.27410	−0.570373
$122$	−2.82612	−0.255865
$123$	0	0
$124$	0.301391	0.0270657
$125$	−3.96567	−0.354700
$126$	0	0
$127$	0.0188127	0.00166936	0.000834680	−	1.00000i	$-0.499734\pi$
$127$	0.0188127	0.00166936	0.000834680		1.00000i	$0.499734\pi$
$128$	5.26909	0.465726
$129$	0	0
$130$	2.88445	0.252983
$131$	5.31238	0.464145	0.232072	−	0.972699i	$-0.425449\pi$
$131$	5.31238	0.464145	0.232072	+	0.972699i	$0.425449\pi$
$132$	0	0
$133$	−8.93433	−0.774704
$134$	14.0214	1.21126
$135$	0	0
$136$	−24.4516	−2.09671
$137$	5.05773	0.432111	0.216055	−	0.976381i	$-0.430681\pi$
$137$	5.05773	0.432111	0.216055	+	0.976381i	$0.430681\pi$
$138$	0	0
$139$	−9.89513	−0.839294	−0.419647	−	0.907687i	$-0.637846\pi$
$139$	−9.89513	−0.839294	−0.419647	+	0.907687i	$0.637846\pi$
$140$	0.909021	0.0768263
$141$	0	0
$142$	14.0490	1.17897
$143$	−12.7260	−1.06420
$144$	0	0
$145$	−2.19509	−0.182292
$146$	9.46899	0.783659
$147$	0	0
$148$	−2.96655	−0.243849
$149$	4.84892	0.397239	0.198620	−	0.980077i	$-0.436354\pi$
$149$	4.84892	0.397239	0.198620	+	0.980077i	$0.436354\pi$
$150$	0	0
$151$	0.196098	0.0159583	0.00797913	−	0.999968i	$-0.497460\pi$
$151$	0.196098	0.0159583	0.00797913	+	0.999968i	$0.497460\pi$
$152$	−6.14021	−0.498037
$153$	0	0
$154$	11.8429	0.954328
$155$	−0.240134	−0.0192880
$156$	0	0
$157$	−5.28786	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$157$	−5.28786	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$158$	−17.4590	−1.38896
$159$	0	0
$160$	1.12334	0.0888075
$161$	−3.02877	−0.238701
$162$	0	0
$163$	−13.1799	−1.03233	−0.516165	−	0.856489i	$-0.672641\pi$
$163$	−13.1799	−1.03233	−0.516165	+	0.856489i	$0.672641\pi$
$164$	−0.200825	−0.0156818
$165$	0	0
$166$	8.54684	0.663364
$167$	−24.2342	−1.87530	−0.937651	−	0.347579i	$-0.887004\pi$
$167$	−24.2342	−1.87530	−0.937651	+	0.347579i	$0.887004\pi$
$168$	0	0
$169$	21.2688	1.63606
$170$	3.93339	0.301677
$171$	0	0
$172$	−6.09130	−0.464458
$173$	23.9278	1.81920	0.909599	−	0.415488i	$-0.136389\pi$
$173$	23.9278	1.81920	0.909599	+	0.415488i	$0.136389\pi$
$174$	0	0
$175$	21.5603	1.62980
$176$	5.93937	0.447697
$177$	0	0
$178$	19.7437	1.47986
$179$	8.14397	0.608709	0.304354	−	0.952559i	$-0.401559\pi$
$179$	8.14397	0.608709	0.304354	+	0.952559i	$0.401559\pi$
$180$	0	0
$181$	−22.0482	−1.63883	−0.819414	−	0.573203i	$-0.805701\pi$
$181$	−22.0482	−1.63883	−0.819414	+	0.573203i	$0.805701\pi$
$182$	−31.8908	−2.36390
$183$	0	0
$184$	−2.08156	−0.153454
$185$	2.36360	0.173775
$186$	0	0
$187$	−17.3538	−1.26904
$188$	3.23049	0.235608
$189$	0	0
$190$	0.987739	0.0716582
$191$	−23.0122	−1.66511	−0.832553	−	0.553945i	$-0.813122\pi$
$191$	−23.0122	−1.66511	−0.832553	+	0.553945i	$0.813122\pi$
$192$	0	0
$193$	7.80810	0.562039	0.281020	−	0.959702i	$-0.409327\pi$
$193$	7.80810	0.562039	0.281020	+	0.959702i	$0.409327\pi$
$194$	−10.5147	−0.754908
$195$	0	0
$196$	−6.50856	−0.464897
$197$	6.78508	0.483417	0.241708	−	0.970349i	$-0.422292\pi$
$197$	6.78508	0.483417	0.241708	+	0.970349i	$0.422292\pi$
$198$	0	0
$199$	−9.79051	−0.694031	−0.347015	−	0.937859i	$-0.612805\pi$
$199$	−9.79051	−0.694031	−0.347015	+	0.937859i	$0.612805\pi$
$200$	14.8175	1.04776
$201$	0	0
$202$	13.5295	0.951934
$203$	24.2691	1.70336
$204$	0	0
$205$	0.160008	0.0111754
$206$	−6.98843	−0.486907
$207$	0	0
$208$	−15.9936	−1.10896
$209$	−4.35783	−0.301438
$210$	0	0
$211$	−11.8919	−0.818673	−0.409337	−	0.912383i	$-0.634240\pi$
$211$	−11.8919	−0.818673	−0.409337	+	0.912383i	$0.634240\pi$
$212$	−4.90327	−0.336758
$213$	0	0
$214$	17.0693	1.16683
$215$	4.85325	0.330989
$216$	0	0
$217$	2.65494	0.180229
$218$	−8.61727	−0.583635
$219$	0	0
$220$	0.443387	0.0298931
$221$	46.7307	3.14345
$222$	0	0
$223$	21.0443	1.40923	0.704614	−	0.709591i	$-0.251120\pi$
$223$	21.0443	1.40923	0.704614	+	0.709591i	$0.251120\pi$
$224$	−12.4197	−0.829827
$225$	0	0
$226$	−21.4903	−1.42951
$227$	−16.5049	−1.09547	−0.547734	−	0.836653i	$-0.684509\pi$
$227$	−16.5049	−1.09547	−0.547734	+	0.836653i	$0.684509\pi$
$228$	0	0
$229$	−5.18227	−0.342454	−0.171227	−	0.985232i	$-0.554773\pi$
$229$	−5.18227	−0.342454	−0.171227	+	0.985232i	$0.554773\pi$
$230$	0.334848	0.0220792
$231$	0	0
$232$	16.6792	1.09505
$233$	11.8271	0.774817	0.387409	−	0.921908i	$-0.373370\pi$
$233$	11.8271	0.774817	0.387409	+	0.921908i	$0.373370\pi$
$234$	0	0
$235$	−2.57389	−0.167902
$236$	−1.45300	−0.0945820
$237$	0	0
$238$	−43.4879	−2.81890
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−15.1921	−0.978607	−0.489304	−	0.872113i	$-0.662749\pi$
$241$	−15.1921	−0.978607	−0.489304	+	0.872113i	$0.662749\pi$
$242$	−7.66891	−0.492976
$243$	0	0
$244$	1.16982	0.0748899
$245$	5.18570	0.331302
$246$	0	0
$247$	11.7349	0.746671
$248$	1.82464	0.115865
$249$	0	0
$250$	−4.84728	−0.306569
$251$	−7.88110	−0.497451	−0.248725	−	0.968574i	$-0.580012\pi$
$251$	−7.88110	−0.497451	−0.248725	+	0.968574i	$0.580012\pi$
$252$	0	0
$253$	−1.47732	−0.0928785
$254$	0.0229950	0.00144284
$255$	0	0
$256$	−11.3002	−0.706263
$257$	28.3840	1.77055	0.885273	−	0.465072i	$-0.153972\pi$
$257$	28.3840	1.77055	0.885273	+	0.465072i	$0.153972\pi$
$258$	0	0
$259$	−26.1322	−1.62377
$260$	−1.19396	−0.0740463
$261$	0	0
$262$	6.49339	0.401163
$263$	16.8514	1.03910	0.519552	−	0.854439i	$-0.326099\pi$
$263$	16.8514	1.03910	0.519552	+	0.854439i	$0.326099\pi$
$264$	0	0
$265$	3.90668	0.239986
$266$	−10.9205	−0.669581
$267$	0	0
$268$	−5.80387	−0.354528
$269$	−7.79527	−0.475286	−0.237643	−	0.971353i	$-0.576375\pi$
$269$	−7.79527	−0.475286	−0.237643	+	0.971353i	$0.576375\pi$
$270$	0	0
$271$	1.75917	0.106862	0.0534309	−	0.998572i	$-0.482984\pi$
$271$	1.75917	0.106862	0.0534309	+	0.998572i	$0.482984\pi$
$272$	−21.8098	−1.32241
$273$	0	0
$274$	6.18212	0.373476
$275$	10.5163	0.634157
$276$	0	0
$277$	28.0737	1.68679	0.843394	−	0.537296i	$-0.180554\pi$
$277$	28.0737	1.68679	0.843394	+	0.537296i	$0.180554\pi$
$278$	−12.0949	−0.725407
$279$	0	0
$280$	5.50326	0.328883
$281$	16.8365	1.00438	0.502190	−	0.864758i	$-0.332528\pi$
$281$	16.8365	1.00438	0.502190	+	0.864758i	$0.332528\pi$
$282$	0	0
$283$	14.3487	0.852941	0.426470	−	0.904502i	$-0.359757\pi$
$283$	14.3487	0.852941	0.426470	+	0.904502i	$0.359757\pi$
$284$	−5.81530	−0.345075
$285$	0	0
$286$	−15.5551	−0.919795
$287$	−1.76906	−0.104424
$288$	0	0
$289$	46.7244	2.74849
$290$	−2.68309	−0.157556
$291$	0	0
$292$	−3.91950	−0.229371
$293$	13.4805	0.787538	0.393769	−	0.919209i	$-0.371171\pi$
$293$	13.4805	0.787538	0.393769	+	0.919209i	$0.371171\pi$
$294$	0	0
$295$	1.15768	0.0674025
$296$	−17.9596	−1.04388
$297$	0	0
$298$	5.92690	0.343336
$299$	3.97816	0.230063
$300$	0	0
$301$	−53.6580	−3.09280
$302$	0.239693	0.0137928
$303$	0	0
$304$	−5.47680	−0.314116
$305$	−0.932052	−0.0533692
$306$	0	0
$307$	−14.9529	−0.853408	−0.426704	−	0.904391i	$-0.640325\pi$
$307$	−14.9529	−0.853408	−0.426704	+	0.904391i	$0.640325\pi$
$308$	−4.90213	−0.279325
$309$	0	0
$310$	−0.293518	−0.0166707
$311$	−11.1362	−0.631478	−0.315739	−	0.948846i	$-0.602252\pi$
$311$	−11.1362	−0.631478	−0.315739	+	0.948846i	$0.602252\pi$
$312$	0	0
$313$	6.43716	0.363850	0.181925	−	0.983312i	$-0.441767\pi$
$313$	6.43716	0.363850	0.181925	+	0.983312i	$0.441767\pi$
$314$	−6.46342	−0.364752
$315$	0	0
$316$	7.22681	0.406540
$317$	16.0732	0.902763	0.451382	−	0.892331i	$-0.350931\pi$
$317$	16.0732	0.902763	0.451382	+	0.892331i	$0.350931\pi$
$318$	0	0
$319$	11.8376	0.662778
$320$	3.57579	0.199893
$321$	0	0
$322$	−3.70211	−0.206310
$323$	16.0023	0.890390
$324$	0	0
$325$	−28.3185	−1.57083
$326$	−16.1100	−0.892248
$327$	0	0
$328$	−1.21581	−0.0671317
$329$	28.4572	1.56890
$330$	0	0
$331$	11.3812	0.625567	0.312784	−	0.949824i	$-0.398739\pi$
$331$	11.3812	0.625567	0.312784	+	0.949824i	$0.398739\pi$
$332$	−3.53779	−0.194162
$333$	0	0
$334$	−29.6218	−1.62083
$335$	4.62424	0.252649
$336$	0	0
$337$	6.74602	0.367479	0.183740	−	0.982975i	$-0.441180\pi$
$337$	6.74602	0.367479	0.183740	+	0.982975i	$0.441180\pi$
$338$	25.9971	1.41406
$339$	0	0
$340$	−1.62815	−0.0882986
$341$	1.29498	0.0701272
$342$	0	0
$343$	−26.1353	−1.41117
$344$	−36.8770	−1.98828
$345$	0	0
$346$	29.2473	1.57234
$347$	−6.94503	−0.372829	−0.186414	−	0.982471i	$-0.559687\pi$
$347$	−6.94503	−0.372829	−0.186414	+	0.982471i	$0.559687\pi$
$348$	0	0
$349$	15.7152	0.841217	0.420609	−	0.907242i	$-0.361817\pi$
$349$	15.7152	0.841217	0.420609	+	0.907242i	$0.361817\pi$
$350$	26.3534	1.40865
$351$	0	0
$352$	−6.05787	−0.322886
$353$	−11.7091	−0.623215	−0.311607	−	0.950211i	$-0.600867\pi$
$353$	−11.7091	−0.623215	−0.311607	+	0.950211i	$0.600867\pi$
$354$	0	0
$355$	4.63334	0.245912
$356$	−8.17253	−0.433143
$357$	0	0
$358$	9.95448	0.526110
$359$	20.0346	1.05738	0.528692	−	0.848814i	$-0.322683\pi$
$359$	20.0346	1.05738	0.528692	+	0.848814i	$0.322683\pi$
$360$	0	0
$361$	−14.9816	−0.788503
$362$	−26.9497	−1.41645
$363$	0	0
$364$	13.2005	0.691897
$365$	3.12286	0.163458
$366$	0	0
$367$	−20.9033	−1.09114	−0.545572	−	0.838064i	$-0.683688\pi$
$367$	−20.9033	−1.09114	−0.545572	+	0.838064i	$0.683688\pi$
$368$	−1.85666	−0.0967848
$369$	0	0
$370$	2.88906	0.150195
$371$	−43.1927	−2.24245
$372$	0	0
$373$	2.89285	0.149786	0.0748930	−	0.997192i	$-0.476138\pi$
$373$	2.89285	0.149786	0.0748930	+	0.997192i	$0.476138\pi$
$374$	−21.2118	−1.09684
$375$	0	0
$376$	19.5575	1.00860
$377$	−31.8765	−1.64172
$378$	0	0
$379$	23.5553	1.20995	0.604977	−	0.796243i	$-0.293182\pi$
$379$	23.5553	1.20995	0.604977	+	0.796243i	$0.293182\pi$
$380$	−0.408855	−0.0209738
$381$	0	0
$382$	−28.1281	−1.43916
$383$	8.52161	0.435434	0.217717	−	0.976012i	$-0.430139\pi$
$383$	8.52161	0.435434	0.217717	+	0.976012i	$0.430139\pi$
$384$	0	0
$385$	3.90577	0.199057
$386$	9.54394	0.485774
$387$	0	0
$388$	4.35233	0.220956
$389$	−20.0809	−1.01814	−0.509072	−	0.860724i	$-0.670011\pi$
$389$	−20.0809	−1.01814	−0.509072	+	0.860724i	$0.670011\pi$
$390$	0	0
$391$	5.42483	0.274345
$392$	−39.4032	−1.99016
$393$	0	0
$394$	8.29349	0.417820
$395$	−5.75796	−0.289714
$396$	0	0
$397$	25.3899	1.27428	0.637141	−	0.770748i	$-0.280117\pi$
$397$	25.3899	1.27428	0.637141	+	0.770748i	$0.280117\pi$
$398$	−11.9671	−0.599855
$399$	0	0
$400$	13.2166	0.660828
$401$	−1.99676	−0.0997136	−0.0498568	−	0.998756i	$-0.515876\pi$
$401$	−1.99676	−0.0997136	−0.0498568	+	0.998756i	$0.515876\pi$
$402$	0	0
$403$	−3.48715	−0.173707
$404$	−5.60027	−0.278624
$405$	0	0
$406$	29.6645	1.47222
$407$	−12.7463	−0.631811
$408$	0	0
$409$	−23.0181	−1.13817	−0.569085	−	0.822279i	$-0.692702\pi$
$409$	−23.0181	−1.13817	−0.569085	+	0.822279i	$0.692702\pi$
$410$	0.195579	0.00965898
$411$	0	0
$412$	2.89272	0.142514
$413$	−12.7994	−0.629816
$414$	0	0
$415$	2.81874	0.138366
$416$	16.3128	0.799799
$417$	0	0
$418$	−5.32664	−0.260534
$419$	−3.68635	−0.180090	−0.0900450	−	0.995938i	$-0.528701\pi$
$419$	−3.68635	−0.180090	−0.0900450	+	0.995938i	$0.528701\pi$
$420$	0	0
$421$	19.3486	0.942992	0.471496	−	0.881868i	$-0.343714\pi$
$421$	19.3486	0.942992	0.471496	+	0.881868i	$0.343714\pi$
$422$	−14.5356	−0.707584
$423$	0	0
$424$	−29.6846	−1.44161
$425$	−38.6165	−1.87318
$426$	0	0
$427$	10.3049	0.498687
$428$	−7.06549	−0.341523
$429$	0	0
$430$	5.93219	0.286076
$431$	−10.1280	−0.487848	−0.243924	−	0.969794i	$-0.578435\pi$
$431$	−10.1280	−0.487848	−0.243924	+	0.969794i	$0.578435\pi$
$432$	0	0
$433$	−34.6573	−1.66552	−0.832761	−	0.553632i	$-0.813241\pi$
$433$	−34.6573	−1.66552	−0.832761	+	0.553632i	$0.813241\pi$
$434$	3.24517	0.155773
$435$	0	0
$436$	3.56695	0.170826
$437$	1.36227	0.0651660
$438$	0	0
$439$	3.02301	0.144280	0.0721402	−	0.997395i	$-0.477017\pi$
$439$	3.02301	0.144280	0.0721402	+	0.997395i	$0.477017\pi$
$440$	2.68429	0.127968
$441$	0	0
$442$	57.1195	2.71690
$443$	−23.1069	−1.09784	−0.548922	−	0.835874i	$-0.684962\pi$
$443$	−23.1069	−1.09784	−0.548922	+	0.835874i	$0.684962\pi$
$444$	0	0
$445$	6.51147	0.308673
$446$	25.7227	1.21800
$447$	0	0
$448$	−39.5343	−1.86782
$449$	13.0483	0.615787	0.307894	−	0.951421i	$-0.400376\pi$
$449$	13.0483	0.615787	0.307894	+	0.951421i	$0.400376\pi$
$450$	0	0
$451$	−0.862882	−0.0406315
$452$	8.89548	0.418408
$453$	0	0
$454$	−20.1741	−0.946819
$455$	−10.5175	−0.493070
$456$	0	0
$457$	−22.9529	−1.07369	−0.536846	−	0.843680i	$-0.680384\pi$
$457$	−22.9529	−1.07369	−0.536846	+	0.843680i	$0.680384\pi$
$458$	−6.33436	−0.295985
$459$	0	0
$460$	−0.138603	−0.00646242
$461$	29.2496	1.36229	0.681145	−	0.732148i	$-0.261482\pi$
$461$	29.2496	1.36229	0.681145	+	0.732148i	$0.261482\pi$
$462$	0	0
$463$	−0.209955	−0.00975742	−0.00487871	−	0.999988i	$-0.501553\pi$
$463$	−0.209955	−0.00975742	−0.00487871	+	0.999988i	$0.501553\pi$
$464$	14.8771	0.690653
$465$	0	0
$466$	14.4564	0.669679
$467$	25.6201	1.18556	0.592778	−	0.805366i	$-0.298031\pi$
$467$	25.6201	1.18556	0.592778	+	0.805366i	$0.298031\pi$
$468$	0	0
$469$	−51.1260	−2.36078
$470$	−3.14610	−0.145119
$471$	0	0
$472$	−8.79651	−0.404892
$473$	−26.1724	−1.20341
$474$	0	0
$475$	−9.69726	−0.444941
$476$	18.0009	0.825072
$477$	0	0
$478$	1.22231	0.0559073
$479$	−31.5384	−1.44103	−0.720513	−	0.693441i	$-0.756094\pi$
$479$	−31.5384	−1.44103	−0.720513	+	0.693441i	$0.756094\pi$
$480$	0	0
$481$	34.3235	1.56502
$482$	−18.5695	−0.845816
$483$	0	0
$484$	3.17439	0.144291
$485$	−3.46772	−0.157461
$486$	0	0
$487$	−13.2957	−0.602486	−0.301243	−	0.953547i	$-0.597402\pi$
$487$	−13.2957	−0.602486	−0.301243	+	0.953547i	$0.597402\pi$
$488$	7.08213	0.320593
$489$	0	0
$490$	6.33855	0.286346
$491$	13.8843	0.626588	0.313294	−	0.949656i	$-0.398568\pi$
$491$	13.8843	0.626588	0.313294	+	0.949656i	$0.398568\pi$
$492$	0	0
$493$	−43.4684	−1.95772
$494$	14.3437	0.645352
$495$	0	0
$496$	1.62749	0.0730767
$497$	−51.2267	−2.29783
$498$	0	0
$499$	7.25368	0.324719	0.162360	−	0.986732i	$-0.448090\pi$
$499$	7.25368	0.324719	0.162360	+	0.986732i	$0.448090\pi$
$500$	2.00644	0.0897305
$501$	0	0
$502$	−9.63317	−0.429949
$503$	6.18887	0.275948	0.137974	−	0.990436i	$-0.455941\pi$
$503$	6.18887	0.275948	0.137974	+	0.990436i	$0.455941\pi$
$504$	0	0
$505$	4.46202	0.198557
$506$	−1.80575	−0.0802754
$507$	0	0
$508$	−0.00951833	−0.000422308 0
$509$	9.55970	0.423726	0.211863	−	0.977299i	$-0.432047\pi$
$509$	9.55970	0.423726	0.211863	+	0.977299i	$0.432047\pi$
$510$	0	0
$511$	−34.5267	−1.52737
$512$	−24.3506	−1.07615
$513$	0	0
$514$	34.6941	1.53029
$515$	−2.30478	−0.101561
$516$	0	0
$517$	13.8804	0.610458
$518$	−31.9417	−1.40344
$519$	0	0
$520$	−7.22830	−0.316982
$521$	38.5824	1.69033	0.845163	−	0.534509i	$-0.179503\pi$
$521$	38.5824	1.69033	0.845163	+	0.534509i	$0.179503\pi$
$522$	0	0
$523$	24.4407	1.06872	0.534359	−	0.845258i	$-0.320553\pi$
$523$	24.4407	1.06872	0.534359	+	0.845258i	$0.320553\pi$
$524$	−2.68781	−0.117417
$525$	0	0
$526$	20.5977	0.898104
$527$	−4.75526	−0.207142
$528$	0	0
$529$	−22.5382	−0.979921
$530$	4.77519	0.207421
$531$	0	0
$532$	4.52034	0.195982
$533$	2.32359	0.100646
$534$	0	0
$535$	5.62943	0.243381
$536$	−35.1369	−1.51768
$537$	0	0
$538$	−9.52825	−0.410792
$539$	−27.9652	−1.20455
$540$	0	0
$541$	7.78761	0.334815	0.167408	−	0.985888i	$-0.446460\pi$
$541$	7.78761	0.334815	0.167408	+	0.985888i	$0.446460\pi$
$542$	2.15025	0.0923612
$543$	0	0
$544$	22.2449	0.953743
$545$	−2.84197	−0.121737
$546$	0	0
$547$	0.826426	0.0353354	0.0176677	−	0.999844i	$-0.494376\pi$
$547$	0.826426	0.0353354	0.0176677	+	0.999844i	$0.494376\pi$
$548$	−2.55897	−0.109314
$549$	0	0
$550$	12.8542	0.548105
$551$	−10.9156	−0.465022
$552$	0	0
$553$	63.6606	2.70712
$554$	34.3149	1.45790
$555$	0	0
$556$	5.00646	0.212321
$557$	27.4842	1.16454	0.582272	−	0.812994i	$-0.302164\pi$
$557$	27.4842	1.16454	0.582272	+	0.812994i	$0.302164\pi$
$558$	0	0
$559$	70.4775	2.98088
$560$	4.90866	0.207429
$561$	0	0
$562$	20.5794	0.868090
$563$	9.93084	0.418535	0.209267	−	0.977858i	$-0.432892\pi$
$563$	9.93084	0.418535	0.209267	+	0.977858i	$0.432892\pi$
$564$	0	0
$565$	−7.08748	−0.298172
$566$	17.5386	0.737201
$567$	0	0
$568$	−35.2061	−1.47722
$569$	35.7909	1.50043	0.750216	−	0.661193i	$-0.229950\pi$
$569$	35.7909	1.50043	0.750216	+	0.661193i	$0.229950\pi$
$570$	0	0
$571$	26.5947	1.11295	0.556476	−	0.830864i	$-0.312153\pi$
$571$	26.5947	1.11295	0.556476	+	0.830864i	$0.312153\pi$
$572$	6.43874	0.269217
$573$	0	0
$574$	−2.16234	−0.0902545
$575$	−3.28741	−0.137094
$576$	0	0
$577$	−18.2595	−0.760154	−0.380077	−	0.924955i	$-0.624102\pi$
$577$	−18.2595	−0.760154	−0.380077	+	0.924955i	$0.624102\pi$
$578$	57.1118	2.37554
$579$	0	0
$580$	1.11061	0.0461156
$581$	−31.1643	−1.29291
$582$	0	0
$583$	−21.0678	−0.872538
$584$	−23.7288	−0.981906
$585$	0	0
$586$	16.4774	0.680673
$587$	33.6391	1.38843	0.694217	−	0.719766i	$-0.255751\pi$
$587$	33.6391	1.38843	0.694217	+	0.719766i	$0.255751\pi$
$588$	0	0
$589$	−1.19413	−0.0492031
$590$	1.41504	0.0582564
$591$	0	0
$592$	−16.0192	−0.658384
$593$	−42.7234	−1.75444	−0.877221	−	0.480088i	$-0.840605\pi$
$593$	−42.7234	−1.75444	−0.877221	+	0.480088i	$0.840605\pi$
$594$	0	0
$595$	−14.3423	−0.587975
$596$	−2.45332	−0.100492
$597$	0	0
$598$	4.86256	0.198845
$599$	39.6329	1.61936	0.809679	−	0.586873i	$-0.199641\pi$
$599$	39.6329	1.61936	0.809679	+	0.586873i	$0.199641\pi$
$600$	0	0
$601$	−0.109685	−0.00447414	−0.00223707	−	0.999997i	$-0.500712\pi$
$601$	−0.109685	−0.00447414	−0.00223707	+	0.999997i	$0.500712\pi$
$602$	−65.5869	−2.67312
$603$	0	0
$604$	−0.0992163	−0.00403705
$605$	−2.52920	−0.102827
$606$	0	0
$607$	24.6085	0.998827	0.499414	−	0.866364i	$-0.333549\pi$
$607$	24.6085	0.998827	0.499414	+	0.866364i	$0.333549\pi$
$608$	5.58607	0.226545
$609$	0	0
$610$	−1.13926	−0.0461273
$611$	−37.3774	−1.51213
$612$	0	0
$613$	−18.4684	−0.745931	−0.372965	−	0.927845i	$-0.621659\pi$
$613$	−18.4684	−0.745931	−0.372965	+	0.927845i	$0.621659\pi$
$614$	−18.2771	−0.737605
$615$	0	0
$616$	−29.6777	−1.19575
$617$	−6.85488	−0.275967	−0.137983	−	0.990435i	$-0.544062\pi$
$617$	−6.85488	−0.275967	−0.137983	+	0.990435i	$0.544062\pi$
$618$	0	0
$619$	23.9096	0.961007	0.480503	−	0.876993i	$-0.340454\pi$
$619$	23.9096	0.961007	0.480503	+	0.876993i	$0.340454\pi$
$620$	0.121496	0.00487940
$621$	0	0
$622$	−13.6120	−0.545790
$623$	−71.9914	−2.88427
$624$	0	0
$625$	22.5889	0.903554
$626$	7.86823	0.314478
$627$	0	0
$628$	2.67540	0.106760
$629$	46.8053	1.86625
$630$	0	0
$631$	30.1531	1.20037	0.600187	−	0.799859i	$-0.295093\pi$
$631$	30.1531	1.20037	0.600187	+	0.799859i	$0.295093\pi$
$632$	43.7514	1.74034
$633$	0	0
$634$	19.6465	0.780263
$635$	0.00758374	0.000300951 0
$636$	0	0
$637$	75.3053	2.98370
$638$	14.4692	0.572842
$639$	0	0
$640$	2.12406	0.0839609
$641$	−3.69224	−0.145835	−0.0729174	−	0.997338i	$-0.523231\pi$
$641$	−3.69224	−0.145835	−0.0729174	+	0.997338i	$0.523231\pi$
$642$	0	0
$643$	41.0966	1.62069	0.810346	−	0.585952i	$-0.199279\pi$
$643$	41.0966	1.62069	0.810346	+	0.585952i	$0.199279\pi$
$644$	1.53241	0.0603855
$645$	0	0
$646$	19.5598	0.769569
$647$	6.20476	0.243934	0.121967	−	0.992534i	$-0.461080\pi$
$647$	6.20476	0.243934	0.121967	+	0.992534i	$0.461080\pi$
$648$	0	0
$649$	−6.24306	−0.245062
$650$	−34.6140	−1.35767
$651$	0	0
$652$	6.66839	0.261154
$653$	−1.87880	−0.0735231	−0.0367615	−	0.999324i	$-0.511704\pi$
$653$	−1.87880	−0.0735231	−0.0367615	+	0.999324i	$0.511704\pi$
$654$	0	0
$655$	2.14151	0.0836758
$656$	−1.08444	−0.0423404
$657$	0	0
$658$	34.7836	1.35601
$659$	36.3588	1.41634	0.708170	−	0.706042i	$-0.249521\pi$
$659$	36.3588	1.41634	0.708170	+	0.706042i	$0.249521\pi$
$660$	0	0
$661$	−48.1205	−1.87167	−0.935836	−	0.352436i	$-0.885353\pi$
$661$	−48.1205	−1.87167	−0.935836	+	0.352436i	$0.885353\pi$
$662$	13.9114	0.540681
$663$	0	0
$664$	−21.4180	−0.831179
$665$	−3.60158	−0.139663
$666$	0	0
$667$	−3.70045	−0.143282
$668$	12.2614	0.474406
$669$	0	0
$670$	5.65226	0.218366
$671$	5.02633	0.194039
$672$	0	0
$673$	17.7655	0.684811	0.342405	−	0.939552i	$-0.388758\pi$
$673$	17.7655	0.684811	0.342405	+	0.939552i	$0.388758\pi$
$674$	8.24575	0.317614
$675$	0	0
$676$	−10.7610	−0.413884
$677$	−18.9540	−0.728463	−0.364232	−	0.931308i	$-0.618668\pi$
$677$	−18.9540	−0.728463	−0.364232	+	0.931308i	$0.618668\pi$
$678$	0	0
$679$	38.3395	1.47133
$680$	−9.85688	−0.377994
$681$	0	0
$682$	1.58287	0.0606113
$683$	−19.7409	−0.755363	−0.377681	−	0.925936i	$-0.623279\pi$
$683$	−19.7409	−0.755363	−0.377681	+	0.925936i	$0.623279\pi$
$684$	0	0
$685$	2.03886	0.0779007
$686$	−31.9455	−1.21968
$687$	0	0
$688$	−32.8927	−1.25402
$689$	56.7317	2.16131
$690$	0	0
$691$	−13.4678	−0.512338	−0.256169	−	0.966632i	$-0.582460\pi$
$691$	−13.4678	−0.512338	−0.256169	+	0.966632i	$0.582460\pi$
$692$	−12.1063	−0.460213
$693$	0	0
$694$	−8.48900	−0.322238
$695$	−3.98890	−0.151308
$696$	0	0
$697$	3.16856	0.120018
$698$	19.2089	0.727069
$699$	0	0
$700$	−10.9085	−0.412301
$701$	−36.8430	−1.39154	−0.695770	−	0.718265i	$-0.744937\pi$
$701$	−36.8430	−1.39154	−0.695770	+	0.718265i	$0.744937\pi$
$702$	0	0
$703$	11.7536	0.443295
$704$	−19.2834	−0.726769
$705$	0	0
$706$	−14.3122	−0.538648
$707$	−49.3325	−1.85534
$708$	0	0
$709$	−2.68748	−0.100931	−0.0504653	−	0.998726i	$-0.516070\pi$
$709$	−2.68748	−0.100931	−0.0504653	+	0.998726i	$0.516070\pi$
$710$	5.66339	0.212543
$711$	0	0
$712$	−49.4769	−1.85422
$713$	−0.404813	−0.0151604
$714$	0	0
$715$	−5.13007	−0.191854
$716$	−4.12046	−0.153989
$717$	0	0
$718$	24.4885	0.913903
$719$	18.2224	0.679582	0.339791	−	0.940501i	$-0.389644\pi$
$719$	18.2224	0.679582	0.339791	+	0.940501i	$0.389644\pi$
$720$	0	0
$721$	25.4818	0.948993
$722$	−18.3122	−0.681508
$723$	0	0
$724$	11.1553	0.414584
$725$	26.3416	0.978301
$726$	0	0
$727$	−7.91643	−0.293604	−0.146802	−	0.989166i	$-0.546898\pi$
$727$	−7.91643	−0.293604	−0.146802	+	0.989166i	$0.546898\pi$
$728$	79.9167	2.96191
$729$	0	0
$730$	3.81711	0.141278
$731$	96.1068	3.55464
$732$	0	0
$733$	1.87435	0.0692307	0.0346154	−	0.999401i	$-0.488979\pi$
$733$	1.87435	0.0692307	0.0346154	+	0.999401i	$0.488979\pi$
$734$	−25.5504	−0.943082
$735$	0	0
$736$	1.89370	0.0698027
$737$	−24.9374	−0.918580
$738$	0	0
$739$	−27.7482	−1.02074	−0.510368	−	0.859956i	$-0.670491\pi$
$739$	−27.7482	−1.02074	−0.510368	+	0.859956i	$0.670491\pi$
$740$	−1.19587	−0.0439610
$741$	0	0
$742$	−52.7949	−1.93816
$743$	−44.2764	−1.62434	−0.812171	−	0.583420i	$-0.801714\pi$
$743$	−44.2764	−1.62434	−0.812171	+	0.583420i	$0.801714\pi$
$744$	0	0
$745$	1.95469	0.0716141
$746$	3.53597	0.129461
$747$	0	0
$748$	8.78020	0.321036
$749$	−62.2395	−2.27418
$750$	0	0
$751$	−38.6448	−1.41017	−0.705084	−	0.709124i	$-0.749091\pi$
$751$	−38.6448	−1.41017	−0.705084	+	0.709124i	$0.749091\pi$
$752$	17.4444	0.636134
$753$	0	0
$754$	−38.9630	−1.41895
$755$	0.0790506	0.00287695
$756$	0	0
$757$	−24.7185	−0.898408	−0.449204	−	0.893429i	$-0.648292\pi$
$757$	−24.7185	−0.898408	−0.449204	+	0.893429i	$0.648292\pi$
$758$	28.7919	1.04577
$759$	0	0
$760$	−2.47523	−0.0897859
$761$	−48.9136	−1.77312	−0.886559	−	0.462616i	$-0.846911\pi$
$761$	−48.9136	−1.77312	−0.886559	+	0.462616i	$0.846911\pi$
$762$	0	0
$763$	31.4211	1.13752
$764$	11.6431	0.421232
$765$	0	0
$766$	10.4161	0.376348
$767$	16.8114	0.607026
$768$	0	0
$769$	−44.2367	−1.59522	−0.797609	−	0.603175i	$-0.793902\pi$
$769$	−44.2367	−1.59522	−0.797609	+	0.603175i	$0.793902\pi$
$770$	4.77408	0.172046
$771$	0	0
$772$	−3.95052	−0.142182
$773$	1.82530	0.0656513	0.0328257	−	0.999461i	$-0.489549\pi$
$773$	1.82530	0.0656513	0.0328257	+	0.999461i	$0.489549\pi$
$774$	0	0
$775$	2.88166	0.103512
$776$	26.3492	0.945882
$777$	0	0
$778$	−24.5452	−0.879988
$779$	0.795679	0.0285082
$780$	0	0
$781$	−24.9865	−0.894087
$782$	6.63084	0.237118
$783$	0	0
$784$	−35.1458	−1.25521
$785$	−2.13163	−0.0760811
$786$	0	0
$787$	−4.69705	−0.167432	−0.0837159	−	0.996490i	$-0.526679\pi$
$787$	−4.69705	−0.167432	−0.0837159	+	0.996490i	$0.526679\pi$
$788$	−3.43292	−0.122293
$789$	0	0
$790$	−7.03803	−0.250402
$791$	78.3598	2.78615
$792$	0	0
$793$	−13.5350	−0.480642
$794$	31.0344	1.10137
$795$	0	0
$796$	4.95353	0.175573
$797$	−27.0924	−0.959663	−0.479831	−	0.877361i	$-0.659302\pi$
$797$	−27.0924	−0.959663	−0.479831	+	0.877361i	$0.659302\pi$
$798$	0	0
$799$	−50.9697	−1.80318
$800$	−13.4803	−0.476600
$801$	0	0
$802$	−2.44067	−0.0861830
$803$	−16.8408	−0.594300
$804$	0	0
$805$	−1.22095	−0.0430329
$806$	−4.26239	−0.150136
$807$	0	0
$808$	−33.9043	−1.19275
$809$	−25.4431	−0.894533	−0.447266	−	0.894401i	$-0.647602\pi$
$809$	−25.4431	−0.894533	−0.447266	+	0.894401i	$0.647602\pi$
$810$	0	0
$811$	−1.52467	−0.0535386	−0.0267693	−	0.999642i	$-0.508522\pi$
$811$	−1.52467	−0.0535386	−0.0267693	+	0.999642i	$0.508522\pi$
$812$	−12.2790	−0.430909
$813$	0	0
$814$	−15.5800	−0.546078
$815$	−5.31305	−0.186108
$816$	0	0
$817$	24.1340	0.844343
$818$	−28.1353	−0.983727
$819$	0	0
$820$	−0.0809562	−0.00282711
$821$	25.4471	0.888111	0.444056	−	0.895999i	$-0.353539\pi$
$821$	25.4471	0.888111	0.444056	+	0.895999i	$0.353539\pi$
$822$	0	0
$823$	49.2594	1.71708	0.858538	−	0.512750i	$-0.171373\pi$
$823$	49.2594	1.71708	0.858538	+	0.512750i	$0.171373\pi$
$824$	17.5127	0.610083
$825$	0	0
$826$	−15.6448	−0.544354
$827$	23.6493	0.822367	0.411183	−	0.911553i	$-0.365116\pi$
$827$	23.6493	0.822367	0.411183	+	0.911553i	$0.365116\pi$
$828$	0	0
$829$	23.2134	0.806234	0.403117	−	0.915149i	$-0.367927\pi$
$829$	23.2134	0.806234	0.403117	+	0.915149i	$0.367927\pi$
$830$	3.44538	0.119591
$831$	0	0
$832$	51.9266	1.80023
$833$	102.690	3.55800
$834$	0	0
$835$	−9.76924	−0.338079
$836$	2.20485	0.0762565
$837$	0	0
$838$	−4.50587	−0.155653
$839$	22.9974	0.793957	0.396978	−	0.917828i	$-0.370059\pi$
$839$	22.9974	0.793957	0.396978	+	0.917828i	$0.370059\pi$
$840$	0	0
$841$	0.651179	0.0224545
$842$	23.6500	0.815033
$843$	0	0
$844$	6.01674	0.207105
$845$	8.57382	0.294948
$846$	0	0
$847$	27.9631	0.960822
$848$	−26.4774	−0.909236
$849$	0	0
$850$	−47.2015	−1.61900
$851$	3.98452	0.136587
$852$	0	0
$853$	5.94006	0.203384	0.101692	−	0.994816i	$-0.467574\pi$
$853$	5.94006	0.203384	0.101692	+	0.994816i	$0.467574\pi$
$854$	12.5958	0.431018
$855$	0	0
$856$	−42.7748	−1.46201
$857$	−45.2128	−1.54444	−0.772219	−	0.635356i	$-0.780853\pi$
$857$	−45.2128	−1.54444	−0.772219	+	0.635356i	$0.780853\pi$
$858$	0	0
$859$	37.9258	1.29401	0.647006	−	0.762485i	$-0.276021\pi$
$859$	37.9258	1.29401	0.647006	+	0.762485i	$0.276021\pi$
$860$	−2.45551	−0.0837322
$861$	0	0
$862$	−12.3796	−0.421650
$863$	−0.597129	−0.0203265	−0.0101633	−	0.999948i	$-0.503235\pi$
$863$	−0.597129	−0.0203265	−0.0101633	+	0.999948i	$0.503235\pi$
$864$	0	0
$865$	9.64571	0.327964
$866$	−42.3620	−1.43952
$867$	0	0
$868$	−1.34327	−0.0455936
$869$	31.0513	1.05334
$870$	0	0
$871$	67.1518	2.27535
$872$	21.5945	0.731281
$873$	0	0
$874$	1.66511	0.0563233
$875$	17.6746	0.597510
$876$	0	0
$877$	10.9382	0.369357	0.184679	−	0.982799i	$-0.440876\pi$
$877$	10.9382	0.369357	0.184679	+	0.982799i	$0.440876\pi$
$878$	3.69506	0.124702
$879$	0	0
$880$	2.39426	0.0807106
$881$	−22.2215	−0.748663	−0.374331	−	0.927295i	$-0.622128\pi$
$881$	−22.2215	−0.748663	−0.374331	+	0.927295i	$0.622128\pi$
$882$	0	0
$883$	−56.4126	−1.89843	−0.949217	−	0.314623i	$-0.898122\pi$
$883$	−56.4126	−1.89843	−0.949217	+	0.314623i	$0.898122\pi$
$884$	−23.6435	−0.795216
$885$	0	0
$886$	−28.2439	−0.948873
$887$	−12.6609	−0.425112	−0.212556	−	0.977149i	$-0.568179\pi$
$887$	−12.6609	−0.425112	−0.212556	+	0.977149i	$0.568179\pi$
$888$	0	0
$889$	−0.0838465	−0.00281212
$890$	7.95905	0.266788
$891$	0	0
$892$	−10.6474	−0.356501
$893$	−12.7993	−0.428314
$894$	0	0
$895$	3.28298	0.109738
$896$	−23.4838	−0.784539
$897$	0	0
$898$	15.9491	0.532228
$899$	3.24371	0.108184
$900$	0	0
$901$	77.3623	2.57731
$902$	−1.05471	−0.0351181
$903$	0	0
$904$	53.8536	1.79115
$905$	−8.88800	−0.295447
$906$	0	0
$907$	−40.0352	−1.32935	−0.664673	−	0.747134i	$-0.731429\pi$
$907$	−40.0352	−1.32935	−0.664673	+	0.747134i	$0.731429\pi$
$908$	8.35068	0.277127
$909$	0	0
$910$	−12.8557	−0.426163
$911$	−17.5878	−0.582708	−0.291354	−	0.956615i	$-0.594106\pi$
$911$	−17.5878	−0.582708	−0.291354	+	0.956615i	$0.594106\pi$
$912$	0	0
$913$	−15.2008	−0.503072
$914$	−28.0556	−0.927998
$915$	0	0
$916$	2.62198	0.0866327
$917$	−23.6768	−0.781876
$918$	0	0
$919$	−15.7906	−0.520883	−0.260442	−	0.965490i	$-0.583868\pi$
$919$	−15.7906	−0.520883	−0.260442	+	0.965490i	$0.583868\pi$
$920$	−0.839112	−0.0276647
$921$	0	0
$922$	35.7522	1.17744
$923$	67.2841	2.21468
$924$	0	0
$925$	−28.3637	−0.932593
$926$	−0.256630	−0.00843339
$927$	0	0
$928$	−15.1740	−0.498110
$929$	40.0331	1.31344	0.656722	−	0.754133i	$-0.271943\pi$
$929$	40.0331	1.31344	0.656722	+	0.754133i	$0.271943\pi$
$930$	0	0
$931$	25.7872	0.845142
$932$	−5.98393	−0.196010
$933$	0	0
$934$	31.3158	1.02468
$935$	−6.99563	−0.228781
$936$	0	0
$937$	−39.9832	−1.30620	−0.653098	−	0.757274i	$-0.726531\pi$
$937$	−39.9832	−1.30620	−0.653098	+	0.757274i	$0.726531\pi$
$938$	−62.4920	−2.04044
$939$	0	0
$940$	1.30227	0.0424753
$941$	13.1615	0.429052	0.214526	−	0.976718i	$-0.431179\pi$
$941$	13.1615	0.429052	0.214526	+	0.976718i	$0.431179\pi$
$942$	0	0
$943$	0.269738	0.00878389
$944$	−7.84609	−0.255369
$945$	0	0
$946$	−31.9908	−1.04011
$947$	−18.6968	−0.607565	−0.303783	−	0.952741i	$-0.598250\pi$
$947$	−18.6968	−0.607565	−0.303783	+	0.952741i	$0.598250\pi$
$948$	0	0
$949$	45.3493	1.47210
$950$	−11.8531	−0.384565
$951$	0	0
$952$	108.979	3.53202
$953$	4.57308	0.148137	0.0740684	−	0.997253i	$-0.476402\pi$
$953$	4.57308	0.148137	0.0740684	+	0.997253i	$0.476402\pi$
$954$	0	0
$955$	−9.27663	−0.300185
$956$	−0.505952	−0.0163637
$957$	0	0
$958$	−38.5498	−1.24549
$959$	−22.5418	−0.727913
$960$	0	0
$961$	−30.6452	−0.988553
$962$	41.9541	1.35265
$963$	0	0
$964$	7.68646	0.247564
$965$	3.14758	0.101324
$966$	0	0
$967$	−30.1643	−0.970020	−0.485010	−	0.874509i	$-0.661184\pi$
$967$	−30.1643	−0.970020	−0.485010	+	0.874509i	$0.661184\pi$
$968$	19.2179	0.617688
$969$	0	0
$970$	−4.23864	−0.136095
$971$	−29.5541	−0.948437	−0.474219	−	0.880407i	$-0.657269\pi$
$971$	−29.5541	−0.948437	−0.474219	+	0.880407i	$0.657269\pi$
$972$	0	0
$973$	44.1017	1.41383
$974$	−16.2515	−0.520732
$975$	0	0
$976$	6.31694	0.202200
$977$	−25.0126	−0.800223	−0.400112	−	0.916466i	$-0.631029\pi$
$977$	−25.0126	−0.800223	−0.400112	+	0.916466i	$0.631029\pi$
$978$	0	0
$979$	−35.1147	−1.12227
$980$	−2.62372	−0.0838115
$981$	0	0
$982$	16.9709	0.541563
$983$	−6.59634	−0.210391	−0.105195	−	0.994452i	$-0.533547\pi$
$983$	−6.59634	−0.210391	−0.105195	+	0.994452i	$0.533547\pi$
$984$	0	0
$985$	2.73518	0.0871502
$986$	−53.1320	−1.69207
$987$	0	0
$988$	−5.93727	−0.188890
$989$	8.18153	0.260157
$990$	0	0
$991$	−5.17310	−0.164329	−0.0821645	−	0.996619i	$-0.526183\pi$
$991$	−5.17310	−0.164329	−0.0821645	+	0.996619i	$0.526183\pi$
$992$	−1.65997	−0.0527040
$993$	0	0
$994$	−62.6150	−1.98603
$995$	−3.94673	−0.125120
$996$	0	0
$997$	−32.1810	−1.01918	−0.509591	−	0.860417i	$-0.670203\pi$
$997$	−32.1810	−1.01918	−0.509591	+	0.860417i	$0.670203\pi$
$998$	8.86627	0.280657
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.14	yes	20
3.2	odd	2		2151.2.a.j.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.7	✓	20	3.2	odd	2
2151.2.a.k.1.14	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22231 q^{2} -0.505952 q^{4} +0.403117 q^{5} -4.45690 q^{7} -3.06306 q^{8} +O(q^{10})\)	\(q+1.22231 q^{2} -0.505952 q^{4} +0.403117 q^{5} -4.45690 q^{7} -3.06306 q^{8} +0.492736 q^{10} -2.17391 q^{11} +5.85396 q^{13} -5.44773 q^{14} -2.73211 q^{16} +7.98275 q^{17} +2.00460 q^{19} -0.203958 q^{20} -2.65720 q^{22} +0.679568 q^{23} -4.83750 q^{25} +7.15536 q^{26} +2.25498 q^{28} -5.44529 q^{29} -0.595692 q^{31} +2.78662 q^{32} +9.75742 q^{34} -1.79666 q^{35} +5.86330 q^{37} +2.45025 q^{38} -1.23477 q^{40} +0.396926 q^{41} +12.0393 q^{43} +1.09990 q^{44} +0.830645 q^{46} -6.38497 q^{47} +12.8640 q^{49} -5.91293 q^{50} -2.96182 q^{52} +9.69118 q^{53} -0.876342 q^{55} +13.6518 q^{56} -6.65585 q^{58} +2.87181 q^{59} -2.31211 q^{61} -0.728121 q^{62} +8.87034 q^{64} +2.35983 q^{65} +11.4712 q^{67} -4.03889 q^{68} -2.19608 q^{70} +11.4938 q^{71} +7.74678 q^{73} +7.16679 q^{74} -1.01423 q^{76} +9.68892 q^{77} -14.2836 q^{79} -1.10136 q^{80} +0.485167 q^{82} +6.99235 q^{83} +3.21799 q^{85} +14.7158 q^{86} +6.65882 q^{88} +16.1528 q^{89} -26.0905 q^{91} -0.343829 q^{92} -7.80444 q^{94} +0.808091 q^{95} -8.60226 q^{97} +15.7238 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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